Question

Alice knows that 70% of the creatures in Wonderland are anthropomorphic animals. She randomly invites several of Wonderland’s inhabitants to a tea party. Depending on which inhabitants of Wonderland are chosen, the proportion of animals invited may vary. If Alice samples 120 inhabitants, what is the probability that more than 80% of the sampled inhabitants are animals? Give your answer to four decimal places.

Answer 1

Answer: 0.0084

Step-by-step explanation:

Let p be the population proportion of the creatures in Wonderland are anthropomorphic animals.

As per given, p = 0.70

Sample size : n=120

Let $\hat{p}$ be the sample proportion of the creatures are anthropomorphic animals.

Now, required probability :  

$P(\hat{p}>0.80)=P(\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}>\dfrac{0.80-0.70}{\sqrt{\dfrac{0.70\times (1-0.70)}{120}}})\\\\P(\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}>\dfrac{0.10}{\sqrt{\dfrac{0.70\times 30}{120}}})\\\\ =P(Z>\dfrac{0.10}{\sqrt{0.00175}})\ \ \ \ \ [Z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}]\\\\ =P(Z>\dfrac{0.10}{0.04183})\\\\\=P(Z>2.39)\\\\=1-P(Z\leq2.39)\\\\=1- 0.9916= 0.0084$

Hence, the probability that more than 80% of the sampled inhabitants are animals= 0.0084